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Conditions for sides to be inscribed in a circle

Source: IMO LongList 1988, Mongolia 3, Problem 53 of ILL

November 3, 2005
geometry unsolvedgeometry

Problem Statement

Given nn points A1,A2,…,An,A_1, A_2, \ldots, A_n, no three collinear, show that the nn- gon A1A2…An,A_1 A_2 \ldots A_n, is inscribed in a circle if and only if A1A2⋅A3An⋅…⋅An−1An+A2A3⋅A4An⋅…An−1An⋅A1An+…A_1 A_2 \cdot A_3 A_n \cdot \ldots \cdot A_{n-1} A_n + A_2 A_3 \cdot A_4 A_n \cdot \ldots A_{n-1} A_n \cdot A_1 A_n + \ldots +An−1An−2⋅A1An⋅…⋅An−3An+ A_{n-1} A_{n-2} \cdot A_1 A_n \cdot \ldots \cdot A_{n-3} A_n =A1An−1⋅A2An⋅…⋅An−2An= A_1 A_{n-1} \cdot A_2 A_n \cdot \ldots \cdot A_{n-2} A_n, where XYXY denotes the length of the segment XY.XY.
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